Some investigation on Hermitian positive-definite solutions of a nonlinear matrix equation

摘要

In this paper, the Hermitian positive-definite solutions of the matrix equation X~s + A~*X~(-t)A = Q are considered. New necessary and sufficient conditions for the equation to have a Hermitian positive-definite solution are derived. In particular, when A is singular, a new estimate of Hermitian positive-definite solutions is obtained. In the end, based on the fixed point theorem, an iterative algorithm for obtaining the positive-definite solutions of the equation with Q = I is discussed. The error estimations are found.